$L_p +L_q$ and $L_p \cap L_q$ are not isomorphic for all $1 \leq p, q \leq \infty$, $p \neq q$

TL;DR

This paper proves that the Banach spaces formed by the sum and intersection of $L_p$ and $L_q$ spaces are only isomorphic when $p$ equals $q$, resolving a specific open question in functional analysis.

Contribution

It establishes the non-isomorphism of $L_p + L_q$ and $L_p igcap L_q$ for all distinct $p$ and $q$, including the case $p=2$ and $q= extinfty$, answering a previously open problem.

Findings

$L_p + L_q$ and $L_p igcap L_q$ are isomorphic iff $p=q$

In particular, $L_2 + L_{ extinfty}$ and $L_2 igcap L_{ extinfty}$ are not isomorphic

The result confirms the non-isomorphism for all $p eq q$, including the special case $p=2$, $q= extinfty$.

Abstract

We prove that if $1 \leq p, q \leq \infty$, then the spaces $L_p +L_q$ and $L_p \cap L_q$ are isomorphic if and only if $p = q$. In particular, $L_2 +L_{\infty}$ and $L_2 \cap L_{\infty}$ are not isomorphic which is an answer to a question formulated in the paper S. V. Astashkin and L. Maligranda, \textit{$L_p + L_{\infty}$ and $L_p \cap L_{\infty}$ are not isomorphic for all $1 \leq p < \infty, p \neq 2$}, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2181--2194.